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zlatps.f(3) LAPACK zlatps.f(3)

zlatps.f -


subroutine zlatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
 
ZLATPS solves a triangular system of equations with the matrix held in packed storage.

ZLATPS solves a triangular system of equations with the matrix held in packed storage.
Purpose:
 ZLATPS solves one of the triangular systems
A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
Parameters:
UPLO
          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular
TRANS
          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b     (No transpose)
          = 'T':  Solve A**T * x = s*b  (Transpose)
          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)
DIAG
          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular
NORMIN
          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.
N
          N is INTEGER
          The order of the matrix A.  N >= 0.
AP
          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
X
          X is COMPLEX*16 array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.
SCALE
          SCALE is DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.
CNORM
          CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
Further Details:
  A rough bound on x is computed; if that is less than overflow, ZTPSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is
x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end
Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end
We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j
and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).
Definition at line 231 of file zlatps.f.

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